Q: Suppose X~Binomial(10,1/2),Y~Binomial(11,1/2),where X and Y are independent. Then P(X<Y) is
(a) less than 1/2
(b) equal to 1/2
(c) greater than 1/2 but less than equal to 10/11
(d) greater than 10/11

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Show that if X(t( is a real process, then its spectrum h(w0 can be put in the form h9w)=(sigma_X^2/2pie)+1/pie(summation over t=1 to infinity (k(t) cos(wt)) where k(.) is the auotocovariance function and sigma_x^2 =k(0)= variance(X(t)).

Discuss the different forms of Engel curve that are usually employed for fitting to family budget data. In such fitting, how would you tackle the following complications?
(i) Household expenditure on a particular item depends, besides depending on income, on the number of persons per family.
(ii) Consumption of families of the same size differs because of varying age and sex consumption.

Q: 8 (C) Define Gini Coefficient.
The following table presents data on distribution of personal income by decile groups of households for each group separately for both rural and urban sectors of a country for the year 2010.
On a graph paper, draw the Lorenz curve for each sector. Also, compute the Gini coefficient for each sector. Hence compare the income inequality of two sectors.
HOUSEHOLD Percentage share in total income
Rural Urban
0-10 3.0 2.3
10-20 4.4 3.3
20-30 5.4 4.1
30-40 6.4 4.8
40-50 7.4 5.8
50-60 8.3 7.0
60-70 9.6 8.7
70-80 10.9 10.0
80-90 13.8 14.0
90-100 30.8 40.0

DEMOGRAPHY & VITAL STATISTICS
INDIAN STATISTICAL SERVICE 2022
PREVIOUS YEAR QUESTION PAPER SOLUTION

Q: The following part of a life table contains four typographical errors, each of a single digit. Find the errors and correct them:
x l_x (_5^)d_x (_5^)q_x (_5^)L_x T_x e_x^0
20 95772 857 0.008953 476495 4922814 51.553
25 94265 699 0.007368 472566 4456319 47.975
30 94166 800 0.008496 468895 3983754 42.306

Solution:
The corrected typographical errors are underlined as follows:
T_20= 4932814
l_25= 94865
e_25^0= 46.975
T_30= 3983753
There are lots of errors in this table but single digit errors are as above under general conditions.

Note: This question is directly from a research paper we have given answers from there directly. Explanation can’t be provided as it can’t be ascertained which data is correct and which one is incorrect.

Q: What is non-orthogonal data? What is the impact of this in the regular analysis of data? How could such analysis be misleading?

2017 ISS 12(a)
Q: It is desired to use a randomised block design with 4 blocks of size 6, each for testing the effects of five treatments A, B, C, D and E. In each block treatments B, C, D and E are replicated once each while treatment A is replicated twice to ensure more precise estimation and testing for A.
Using a suitable model, give expressions for different sums of squares and write down the ANOVA table.
Discuss the procedure of testing equality of all treatment effects, as also for equality of effects of A and B, if hypothesis of equality of all treatment effects is rejected.

2017 ISS
Q12(d): A 2^3 experiment is proposed to be carried out in 2 incomplete blocks of size 4 each per replicate, retaining full information on all main effects, and sacrificing equal amount of information on the remaining four factorial effects each. Suggest a scheme of partial confounding ensuring the above, and give the intrablock subgroup in each replicate. Discuss the method of analysis for the data.

2019 ISS
Q: 11(A)
Assuming 'p' as a prime number, consider
L_j [■(0&1&2&…&p-1@j&j+1&j+2&…&j+p-1@2j&2j+1&2j+2&…&2j+p-1@…&…&…&…&…@(p-1)j&(p-1)j+1&(p-1)j+2&…&(p-1)j+p-1)]
and prove that L_j is a Latin square.

2019 ISS 12(C)
Q: A 6×6 LSD was used to compare 6 different types of legume. Yields (in 10 gm units) are given in the following table. Two figures are missing. Analyse the data to find if there is any difference among legumes.
B
220 F
98 D
149 A
92 E
282 C
160
A
74 E
238 B
. C
228 F
48 D
168
D
188 C
279 F
118 E
278 B
176 A
.
E
295 B
222 A
64 D
104 C
213 F
163
C
187 D
90 E
242 F
96 A
66 B
188
F
90 A
124 C
195 B
109 D
79 E
211

2019 12d Q: Distinguish between split and strip plot designs and outline the analysis of a split plot designs.

ISS PREVIOUS YEAR SOLUTION
2020 Q: No. 11(a) (10 marks question)
Q: What are randomisation and replication in experimental design? How are these principles useful?

Given X'X={{10,0,0}{0,5,0}{0,0,10}}, X'Y={{12,20}{20,10}{30,20}} estimate the model y_1t=beta_12y_2t+gamma_11 X_(1t)+gamma _(12) X_(2t)+u_(1t) , y_(2t)=beta_(21)y_(1t)+gamma_(23) X_(3t) +u_(2t) using 2sls method

9(a) Q: In the ISS examination of a certain year, the distribution of scores in statistics Papers I, II, III and IV is found to follow a 4 variate normal distribution with known parameters. Indicate how you would estimate the percentage of candidates in that examination faring better (i) in papers I and II compared to papers III and IV, both in the aggregate, and (ii) in paper I compared to paper IV.

ISS PREVIOUS YEAR QUESTION PAPER -2017 SOLUTION Q. Number: 9(b)
Q: In multivariate analysis, often the issue of handling too many variables poses a big problem. How would you propose to solve the problem? Indicate the procedure.

ISS PREVIOUS YEAR QUESTION PAPER SOLUTION
2016 SET A Question no: 64
Q: Consider the following data:
x 1.35 1.36 1.37 1.38
〖log〗_10⁡x 0.1303 0.1335 0.1367 0.1399

What is the value of x when x=10 〖log〗_10⁡xand computed using Stirling’s formula?
(a) 1.3609
(b) 1.3709
(c) 1.3809
(d) 1.3909

The function F(x)= 0;x<0, x/2 ;0<=x<1 (1-alpha)x+2alpha-1;1<=x<2, 1 x>=2 is a distribution function iff

The function of F(x)= 0, x<0, x/3 if 0<=x<=3, (1-alpha)x+3alpha-2 if 2<=x<3, 1, x>=3 is the distribution function of a continous random variable

Suppose f is a three time differentiable function on {0,2}. The integral of [0,2] f(x) dx is computed using trapezoidal rule partitioning [0,2] into 10 equal sub-intervals. The error is -1/150f''(t) for some t belong to t belong to [0,2]

Consider the following data: x:{0,1.0,1.5,2.0} f(x):{1.1, 2.4, 5.7, 8.1}. Let I=intergration of [0,2] f(x) dx. Application of combination of Trapezoidal rule on[0,1] and simpson's one-third rulr on[1,2] gives the value of I as: (a) 7.30 (b) 5.55 (c) 3.65 (d) 3.33

Q:2(C) Consider the Population U={1,2,3,4,5}. For a Sampling design, probabilities of sample drawn are P_r {(1,2)}=0.2,P_r {(2,3,4)}=0.1,P_r {(1,4,5)}=0.3,P_r {(2,4,5)}=0.2,P_r {(1,2,3,4)}=0.2. Calculate the first order inclusion probabilities. Obtain estimates ∑_(i∈S)▒(l_i y_i)/π_i of the parametric function ∑_(i=1)^N▒〖l_i y_i 〗 where l_i’s are known real numbers, not all zero.

Consider a population U={ u_1,u_2,u_3,u_4}. The values of the study variable are y(u_i )=i, i=1(1)4.
A sample of size 2 is drawn from the population with (i) SRSWOR (ii) SRSWR. Calculate V_1=Var(y ̅│SRSWOR) and V_2=Var(y ̅│SRSWR) where y ̅ denotes the sample mean.
A linear systematic sample of size 2 is drawn when the units of the population are arranged as follows:
Arrangement 1: u_1,u_4,u_2,u_3
Arrangement 2: u_1,u_2,u_4,u_3
Calculate Var(y ̅│arrangement 1)=V_3 and Var(y ̅│arrangement 2)=V_4.
Show that V_4<V_1<V_2<V_3.

Explain the steps of Hansen- Hurwitz technique of subsampling of non-respondents for estimating the population mean. or "Discuss and derive the Technique given by hansen and hurwitz for non-response"

The ratio of the sizes and the standard deviations of the two strata are 11 and 9 respectively. The ratio of the sample sizes for a general allocation is n_1/n_2 and that for the Neyman allocation n_1'/n_2', and the ratio of the two ratios is 2. If the objective is to estimate the population mean by taking with replacement random samples from within strata, compare the precision of the two allocations.

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consider the simple linear regression model

Q3(c) .A village has five orchards, containing 15, 30, 25, 10 and 20 trees respectively. If the yields (in 10 kg) of these 5 orchards are 18, 35, 29, 12, and 24 respectively and selecting sample of two units at 2nd and 4th position, estimate the total production of five orchards along with standard error using Horvitz-Thompson estimator.

ISS PREVIOUS YEAR SOLUTION 2022
Q 1 (a): Consider a population of size N. Let S1 be a simple random sample of size n1 drawn without replacement. Another simple random sample S2 of size n2 was also drawn without replacement from the remaining population.
Find the probability of obtaining the combined sample S1∪S2 from the population.
DefineY ̅ ̂_α=αY ̅ ̂_1+(1-α) Y ̅ ̂_2 ,0<α<1. Show that Y ̅ ̂_α is an unbiased estimator for the population mean. Here Y ̅ ̂_i is the mean of sample S_i ,i=1,2.

Evaluate the ARL to detect a shift in the process average(no change in process variation) by one units of standard deviation in the higher side Assume that the use of x bar chart with subgroup 5 and probability of not detecting the shift is at most 0.05

An improvement in the process has resulted an increase in of Cpk from 0.06 to 0.09. Estimate the reduction in % of non-conforming products. Assuming the following: Process has not changed, Cp = Cpk and process is in statistical control.

Let X_1 ,...,X_2n be iid random variables fron normal(meu,simga^2) then the value of k for which t_(2n)=(X) =k{(X_1 -X_2)^2+(X_3-X_4)^2+...(X_(2n-1)-X_(2n))^2 is an unbiased estimator of Simga^2 is

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Consider the following for the next two (02) items that follow: Consider the problem of testing H0: theta=theta0 against H1: theta=theta1 for the probability distribution given below: |X| 0|1 |2 |3 |________________|P_theta0(x)|1/4|1/4|1/4|1/4|__|P_theta1^(x)|0.20|0.40|0.25|0.15| The null hypothesis is rejected if x=0 is observed. It is further rejected when tossing of two unbiased coins gives both the heads for X=1. 39. What is the probabilirt of type-1 error of the test? (a) 3/10 (b) 5/16 (c) 7/10 (d) 1/4 (40) What is the probability of type-II error of the test? (a) 3/10 (b) 5/16 (c) 7/10 (d) 1/4

Let a random variable X have the following probability distribution under H0: theta=theta0 against H1: theta=theta1 : X| -1|1 |2 |________|P_theta0_(x)|1/3|1/3|1/3__________|P_theta1_(x)|1/4|1/4|1/2| To test sequentially the hypothesis H0 against H1, it is decided to continue the procedure as long as -(n+1)/2 <S_n <(n+2)/2 where S_n=Sum_(i=1 to n) X_i. Then the probability under H_1 that the procedure will terminate with the second observation, is: (a) 1/9 (b) 1/8 (c) 1/3 (d)1/2

If θ ̂_1=X/n and θ ̂_2=1/3 are the estimators of the parameter θ of a binomial population and θ=1/2 , then the values of n for which the mean square error of θ ̂_2 is less than the variance of θ ̂_1, are
(a) 2≤n<7 only
(b) 1≤n≤6 only
(c) 1≤n≤8 only
(d) 1≤n≤4 only

Show that real-valued function
r(h)={█(1,if h=0@ρ,if h=±1@0,otherwise)┤
is a auto-covariance function, if |ρ|<1/2

Q. Define stationary time process and autocovariance function. Show that autocovariance function, denoted by γ(h), is an even function, positive semi-definite and uniformly continous if it is continous at h=0.

State ergodicity propert of a time series process. If {X_t, t belongs o T} is a stationary time process with E(X_t)=meu and Variance(X_t)=sigma^2 for all t, define ergodicity of meu. Also, show that the process whose covariance function gamma(h)tends to 0 as h tends to infinity is ergodic for meu.

Q : Let X_1,X_2,… be i.i.d. random variables with uniform distribution on the interval [0, 1]. Let Y_(n,k) denote the kth order statistics based on the sample X_1,…,X_n (e.g; Y_(n,1)=min⁡{ X_1,…,X_n}). What is the probability that〖 Y〗_(21,7)=Y_(22,7)?
1/3
2/3
7/11
15/21